Numerical solution of water hammer phenomenon by Collocated Discrete Least Squares method

“Water hammer” is one of the phenomena that causes damage in the pipe system and reduces their useful life. Various numerical methods have been used to analyze this problem. In all numerical methods, for calculating the variables that are the velocity and pressure values due to the sudden discontinuity of the flow and motion of the pressure wave along the pipe, the continuum environment of the problem must be discretized in some way. With calculating these aberrations before designing of the structures accurately, appropriate measures can be taken to reduce tensions caused by the water-hammer phenomenon. Background and objectives: The conventional method to numerically solve the differential equations that describe this phenomenon is the method of characteristic lines. In general, in conventional methods where have been developed correctly, such as finite element, finite volume and finite difference, discretization of the spatial domain of the problem is done by gridding. Despite the useful use of these methods in many scientific fields, gridding is a costly and troublesome process, especially on problems with complex boundaries. That is the main motive for the creation of meshless methods. In these methods, the spatial domain of the problem is simply discretized by a number of points. In the present study, for modeling classical water-hammer in a system including valve, pipe and reservoir, a collocated discrete least squares method is used. In the proposed approach implicit Crank-Nicolson method for time discretization is used to provide conditions for problem solving stability. In this method, the velocity and pressure values on the x-t plane are calculated directly from the previous time step simultaneously. This method is quite matrix and the solution process is accomplished including several simple algebraic operations on matrices. In this study, at first this numerical method is described generally then governing equations are calculated and several experiments on water hammer in the form of the problem have been modeled by this method, also the hydraulic analysis of problems and calculation steps for calculating accurate answers are fully described and the results are verified with exact answers and other numerical methods such as MOC method and numerical method used by “Zielk” and the computational average error was estimated to be less than 5% by total squared error criterion. So this method can be considered as a precise, simple, and low-cost numerical method for modeling of water-hammer phenomena. Important properties of the Meshless numerical method included no need to integrate, complete mathematical math operations and meshless space makes it one of the most accurate methods for numerical solution of water hammer phenomenon in the pipe system.